2. Objectives
At the end of this lesson, the learner should be able to
correctly illustrate a probability distribution for a
discrete random variable; and
accurately construct a probability distribution for a
discrete random variable.
3. Essential Questions
What is the importance of constructing a probability
distribution for a random variable in interpreting data?
What does the probability distribution tell us about our
expected or obtained data?
4. Warm Up!
Before we thoroughly probability distribution, let us watch to
revisit what probability is by watching a short clip.
Click the link to access the video. (Note: You may watch the
video until 6:42 only)
Mathantics. “Math Antics – Basic Probability.” YouTube video,
11:28. Posted 15 May 2019. Retrieved 22 June 2019 from
https://www.youtube.com/
watch?v=KzfWUEJjG18&feature=youtu.be
5. Guide Questions
● What is probability and how do we calculate it?
● Why is the sum of the probabilities of any event is 1 and
the probabilities are always between 0 and 1, inclusive?
● In the video, the outcomes in tossing fair coin is either
head or tail. What if you toss two coins; what are the
possible outcomes and what is the probability of obtaining
a head?
6. Learn about It!
Probability Distribution of a Discrete Random
Variables
is a list, a table, a graph, or a formula of probabilities associated with each of its
possible values
1
Example:
In tossing two coins, the possible
outcomes are {𝐻𝐻, 𝐻𝑇, 𝑇𝐻, 𝑇𝑇}. If
the random variable 𝑋 denotes the
number of heads in the outcomes,
then the table shows the
probability distribution.
𝐗 0 1 2
𝐏(𝐗) 1
4
2
4
1
4
7. Learn about It!
Properties of a Probability Distribution:
a. The probability of each outcome is between 0 and 1, inclusive. That is, 0 ≤
𝑃 𝑋 = 𝑥𝑖 ≤ 1.
b. The sum of all the probabilities of the random variable is equal to 1 or 100%.
That is, Σ𝑃 𝑋 = 𝑥𝑖 = 1.
2
8. Try It!
Example 1: Determine whether a distribution is a valid
probability distribution for a discrete random variable 𝑋.
𝑿 1 2 3 4
𝑷(𝑿) 0.32 0.28 0.34 0.06
9. Try It!
Example 1: Determine whether a distribution is a valid
probability distribution for a discrete random variable 𝑋.
Solution:
To determine if the distribution is a valid probability
distribution, we must satisfy the two properties for the
probability distribution of a discrete random variable.
𝑿 1 2 3 4
𝑷(𝑿) 0.32 0.28 0.34 0.06
10. Try It!
Example 1: Determine whether a distribution is a valid
probability distribution for a discrete random variable 𝑋.
Solution:
a. The probability of each outcome is between 0 and 1.
The probabilities 0.32, 0.28, 0.34, and 0.06 are all between 0
and 1.
𝑿 1 2 3 4
𝑷(𝑿) 0.32 0.28 0.34 0.06
11. Try It!
Example 1: Determine whether a distribution is a valid
probability distribution for a discrete random variable 𝑋.
Solution:
b. The sum of all the probabilities of the random variable is
equal to 1 or 100%.
0.32 + 0.28 + 0.34 + 0.06 = 1
Thus, the above distribution is a valid probability distribution
for the discrete random variable 𝑋.
𝑿 1 2 3 4
𝑷(𝑿) 0.32 0.28 0.34 0.06
12. Try It!
Example 2: Construct the probability distribution for the
random variable 𝑋 which pertains to the number of tails in
each outcome when tossing two coins.
13. Try It!
Solution:
To construct the probability distribution for a discrete
random variable 𝑋, we need to determine the possible
outcomes of a random experiment.
In tossing two coins, the possible outcomes are 𝑆 =
{𝐻𝐻, 𝑇𝑇, 𝐻𝑇, 𝑇𝐻) where 𝐻 represents the head and 𝑇
represents the tail.
Example 2: Construct the probability distribution for the
random variable 𝑋 which pertains to the number of tails in
each outcome when tossing two coins.
14. Try It!
Solution:
From the outcomes, we can have the following table:
Based on the table above, the random variable 𝑋 can take the
values of 0, 1, and 2.
Example 2: Construct the probability distribution for the
random variable 𝑋 which pertains to the number of tails in
each outcome when tossing two coins.
Number of Tails Outcomes
0 𝐻𝐻
1 𝐻𝑇, 𝑇𝐻
2 𝑇𝑇
15. Try It!
Solution:
Thus, the probability distribution for the discrete random
variable 𝑋 is
Example 2: Construct the probability distribution for the
random variable 𝑋 which pertains to the number of tails in
each outcome when tossing two coins.
𝑿 0 1 2
𝑷(𝑿) 1
4
2
4
or
1
2
1
4
16. Let’s Practice!
Individual Practice:
1. Determine whether the distribution is a valid probability
distribution for a discrete random variable 𝑋.
𝑿 5 6 7 4
𝑷(𝑿) 0.22 0.35 0.18 0.35
17. Let’s Practice!
Individual Practice:
2. Consider the random experiment of rolling a pair of
tetrahedron dice (whose number of dots are 1 to 4).
Construct the probability distribution for the random
variable 𝑋 which denotes the sum of the numbers in the
two dice.
18. Let’s Practice!
Group Practice: To be done in groups of 3.
A bowl contains five marbles numbered as 0, 2, 4, 6, and 8. If
a random experiment of picking three marbles at a time was
conducted, construct a probability distribution for the
random variable 𝑋 which represents the sum of the numbers
on the marbles.
19. Key Points
Probability Distribution of a Discrete Random
Variables
is a list, a table, a graph, or a formula of probabilities associated with each of its
possible values
1
Properties of a Probability Distribution:
a. The probability of each outcome is between 0 and 1, inclusive. That is, 0 ≤
𝑃 𝑋 = 𝑥𝑖 ≤ 1.
b. The sum of all the probabilities of the random variable is equal to 1 or 100%.
That is, Σ𝑃 𝑋 = 𝑥𝑖 = 1.
2
20. Synthesis
● How do we construct a probability distribution for a
random variable 𝑋?
● In what real-life contexts can you apply the concept of
probability distribution for a random variable?
● How can you find the value of a certain random variable?
